the influence of spiral sampling on surface topography

the influence of spiral sampling on surface topography

KOMISJA BUDOWY MASZYN PAN – ODDZIAŁ W POZNANIU
Vol. 27 nr 2
Archiwum Technologii Maszyn i Automatyzacji
2007
RADOMIR MAJCHROWSKI *
THE INFLUENCE OF SPIRAL SAMPLING ON SURFACE
TOPOGRAPHY PARAMETERS – SIMULATION ANALYSIS ∗∗
In this paper, the author presents the result of simulating for surface measurement based on
spiral grid using Matlab software. Most of the topography surface measurement instruments work
using the parallel profiles to obtain surface texture. The novel stylus based measurement instrument that use the spiral sampling is the answer of the question of faster surface topography measurements method. The presented results of comparing three-dimensional surface parameters obtaining from simulating measurement surface based on a rectangular grid and based on a spiral
grid is the first step to determine and characterize the influence of spiral sampling on surface topography parameters. It can make further possible to implement this method into industry as fast
surface topography measurement method based on stylus profilometer.
Key words: spiral sampling, surface topography measurement
1. INTRODUCTION
First machine for the surface measurement was introduced in 1929 by
Schmalz in Germany, in 1933 Abbott’s profilometer was conceived in USA [5].
Stylus profilometers based on this construction make possible characterize of the
Surface just in two dimensions – surface is represented by a single profile. In
recent years, it became clear that surface topography is the best way to proper
characterisation of solid’s micro-geometry. Most of the topography surface
measurement instruments work using the parallel profiles to obtain surface
texture. Unfortunately, stereo metric topography measurement is very much time
consumption.
Division of Metrology and Measuring Systems of Poznan University have
elaborated the novel stylus based measurement instrument [1, 2, 6, 7] that use
the spiral sampling, as an answer to find faster surface topography
measurements method – Fig. 1 (this project was sponsored by Polish Committee
* Mgr
∗∗ This
inż. – Institute of Mechanical Technology, Poznan University of Technology.
research is sponsored from financial support for science within the frames of PhD
Thesis for 2006-2008 years (2219/T02/2006/31).
72
R. Majchrowski
for Scientific Research). In this way, stereo metric measurements surface is
represented as a set of coordinates of grid points, basing on a grid constructed of
the spiral of Archimedes. Currently, in all of the stylus profilometers, sampling
is based upon a rectangular grid, which is easy to measurement execute and
surface topography analyse. All of three-dimensional surface parameters are
based on a rectangular grid, it is the cause for interpolation spiral grid to
rectangular grid. Among other things this procedure must induce some
difference between surface parameters obtained by spiral and rectangular grid.
Fig. 1. Measurement instrument that use the spiral sampling
Rys. 1. Profilometr wykorzystujący siatkę spiralną
The authors are focused to find the relationship between different spiral
sampling and obtaining surface parameters. It is difficult to repeat surface
measurement on exactly the same area, to eliminate this problem, we created
application program, which simulates surface measurement based on spiral grid
using Matlab software. Despite the fact, that spiral grid method is faster as the
method based on rectangular grid, the surface measurement is very time
consumption. It was the next reason of interesting the simulating way to
preliminary characterise the raising problem. Of course, after that all result must
be experimental confirmed.
2. SIMULATION PROCEDURE
It was use four kind of surface: glass plate after sandblasting and the surface
machined by grinding, electrolytic machining and honing (Fig. 2). The data
points for surface topography were collected by means of a Pherten stylus
profilometer and surface topography was measured using rectangular grid 0.4 ×
0.4 mm (matrix: 268 × 268 points). On this surfaces different spiral grids were
The influence of spiral sampling on surface topography parameters …
73
spread. We change the distance between spiral roll from 25 μm to 1.8 μm (16 to
222 spiral roll). For each spiral the parameters were calculated and saved into
Matlabs’s table. Below the simulation algorithm is introduced, it consists of six
point. All of this was developped using Matlab’s programming language [9].
Fig. 2. Four kind of surface: glass plate after sandblasting and the surface machined by grinding,
electrolytic machining and honing
Rys. 2. Cztery rodzaje powierzchni: płytka szklana po piaskowaniu oraz powierzchnie po szlifowaniu,
obróbce elektroerozyjnej i gładzeniu
I – The first step:
Creating the spiral grid in polar coordinates. This is given by formula 1.
ϕ = r ⋅φ ,
(1)
where: r – radius (max r = 0.2),
ф – angle (max ф = π * max number of spiral rolls).
So created spiral grid was transformed from polar coordinates to Cartesian by
formula 2 and 3. The expression ‘(0.2)’ is added in order to transform the origin
of coordinate to point (0,0) – in polar coordinates the value of radius changes in
range ± 0.2 mm.
x = [r ⋅ cos(φ )] + (0.2) ,
y = [r ⋅ sin(φ )] + (0.2) .
(2)
(3)
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R. Majchrowski
II – The second step:
The spiral grid (in Cartesian coordinates system) is spread on the surfaces.
Each surface (lapping, grinding, abrasive blasting, honing) is mapped at the
same position and the same size.
III – The third step:
Interpolation spiral grid to rectangular grid – it was used Matlab function of
two dimensional triangle-based data interpolation – based on a Delaunay
triangulation of the data. Matlab has alternative interpolation method: linear –
triangle-based linear interpolation, nearest – nearest neighbours interpolation,
cubic – triangle-based cubic interpolation. In simulation algorithm were used
cubic method.
IV – The fourth step:
After interpolation the spiral grid to rectangular grid, the surface topography
on area that is approximate to circle which diameter is 0.4 mm, was obtained.
For further analyzing it is necessary to cut some part of area of surface to get the
surface topography on rectangular area. We set the cutting area as square which
circumscribed circle has 0.4 mm diameter. So, the new area of surface is square:
⎛
⎞
0.4
= 0.282 mm ⎟⎟ .
0.282 × 0.282 mm ⎜⎜ a =
2
⎝
⎠
V – The fifth step:
Calculate three-dimensional surface parameters. It was used parameters that
characterise three major aspects of surface topography [3, 4]: amplitude
property, spatial property, hybrid property (combinations of amplitude and
spacing). In this paper, the authors present just some selected parameters.
As amplitude parameters representative are used:
Sq – root-mean-square deviation of surface, it is a dispersion parameter
defined as the root of the mean square value of the surface departures within the
sampling area. It is given by digital formula (4):
Sq =
1
MN
N
M
∑∑η 2 (i, j ) .
(4)
j =1 i =1
This parameter has a very insensitive meaning in statistics (standard
deviation) and is also a very general and widely used parameter. However due
to the squaring operation, is more sensitive to extreme data values than
parameter Sa.
The next amplitude parameters representative are used:
Sku – kurtosis of surface height distribution, it is a measure of the peaked-ness
or sharpness of the surface height distribution. It is given by digital formula (5):
The influence of spiral sampling on surface topography parameters …
S ku =
1
MNS q4
75
N M
∑∑η 4 (i, j ) .
(5)
j =1 i =1
This parameter characterises the spread of the height distribution.
As spatial parameters the following representatives are used:
Sds – density of summits of the surface, it is number of summits of a unit
sampling area. It is given by formula (6):
S ds =
Number of summits
.
(M − 1) ⋅ (N − 1) ⋅ Δx ⋅ Δy
(6)
There is a problem in definition of the summits. In order to have an objective
definition of a surface summit, research used definition that is based on the
geographical approach to classify the features of the earth relief as a family of
summits, crests etc. This analysis algorithm supposes that every point on a
surface can be identified and classified by analysis of its neighbors. In created
application program to simulate surface measurement based on spiral grid the
authors used the 24 neighbors summit identification, that was presented by
Zahouani and others [9].
As hybrid parameters the following representatives are used:
SΔq – root-mean-square slope of the surface, it is defined as the root men
square value of the surface slope within the sampling area. It is given by digital
formula (7):
S Δq =
2
2
N M ⎡
1
⎛ η ( xi , yi ) − η ( xi −1 , yi ) ⎞ ⎛ η ( xi , yi ) − η ( xi , yi −1 ) ⎞ ⎤ .
⎢
⎜
⎟
+
⎜
⎟
⎜
⎟ ⎥
(M − 1)(N − 1) ∑∑
Δy
Δx
⎠ ⎝
j = 2 i = 2 ⎢⎝
⎠ ⎥⎦
⎣
All of this parameters were calculated two times. The first time for the input
surface, and second time for the simulate surface (spiral grid).
VI – The sixth step:
Store procedure – all of obtained surface parameters were stored into
Matlab’s table to further data analyzing and visualization.
VII – The seventh step:
All six steps were repeated for range from 16 to 222 spiral roll.
In Fig. 3 is shown: creating the spiral grid in polar coordinates; the spiral grid
(in Cartesian coordinates system) is spread on the surfaces and interpolation
spiral grid to rectangular grid.
(7)
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R. Majchrowski
Fig. 3. Visualization of three first steps of simulation algorithm
Rys. 3. Wizualizacja pierwszych trzech kroków algorytmu symulacyjnego
The influence of spiral sampling on surface topography parameters …
77
3. RESULTS AND DISCUSSION
Basing on the result of the test comparison analysis of surface parameters
from rectangular and spiral sampling was performed. To demonstrate in a clear
way the issue, authors have created chart, where correlation is shown between
surface parameters from rectangular and spiral sampling and number of spiral
rolls. So, on axis of ordinates there is the expression: ‘(Sq*/Sq) × 100%’. In this
expression Sq* is the parameter calculated for the input surface, and Sq for the
simulate surface (spiral grid). On axis of abscissas there is number of spiral rolls.
For each parameter the simulating procedure was taken two times. The first time
for 16 to 222 spiral rolls and the second time for 65 to 222 spiral rolls (the step
of change was smaller).
The amplitude parameters
Sq – root-mean-square deviation of surface Fig. 4; this amplitude parameter
stabilize after about 100 spiral rolls (distance between spiral roll – 4 μm) for
grinding and honing surfaces. The abrasive blasting change in all range but after
about 60 spiral rolls (distance – 6.6 μm) its trend became probably linear. The
most visible difference is for lapping surface, its trend grown and after 160 spiral
rolls (distance – 2.4 μm) grown less.
Fig. 4. The result of the test – amplitude parameter Sq. Simulating procedure was taken two times:
for 16 to 222 and for 65 to 222 spiral rolls (the step of change was smaller)
Rys. 4. Wyniki testu – parametr amplitudowy Sq . Symulacja wykonana dwa razy: dla 16÷222 i dla
65÷222 zwojów spirali (krok próbkowania był mniejszy)
For next amplitude parameters Sku (kurtosis of surface height distribution) the
result of simulation are similar like amplitude parameter Sq – Fig. 5.
78
R. Majchrowski
Fig. 5. The result of the test – amplitude parameter Sku. Simulating procedure was taken two times:
for 16 to 222 and for 65 to 222 spiral rolls (the step of change was smaller)
Rys. 5. Wyniki testu – parametr amplitudowy Sku. Symulacja wykonana dwa razy: dla 16÷222
i 65÷222 zwojów spirali (krok próbkowania był mniejszy)
The spatial parameter
Sds – density of summits of the surface Fig. 6; for this spatial parameter the
difference between the parameter calculated for the input surface and for the
simulate surface (spiral grid) has constant probably linear trend. It is remarkable
that this trend is growing for all spiral rolls range and is similar for all surfaces.
Fig. 6. The result of the test – spatial parameter Sds. Simulating procedure was taken two times: for
16 to 222 and for 65 to 222 spiral rolls (the step of change was smaller)
Rys. 6. Wyniki testu – parametr przestrzenny Sds. Symulacja wykonana dwa razy: dla 16÷222
i 65÷222 zwojów spirali (krok próbkowania był mniejszy)
The hybrid parameters
SΔq – root–mean–square slope of the surface; this hybrid parameter is very
susceptible to number of spiral rolls (on the beginning quotient of Sq* and Sq has
The influence of spiral sampling on surface topography parameters …
79
value of 200÷400%), but it stabilize after about 140 spiral rolls (distance
between spiral roll – 2.8 μm). For grinding surface SΔq it stabilize faster as for
others surfaces. It is interesting that for this hybrid parameter the trend is
growing less from the beginning.
Fig. 7. The result of the test – hybrid parameter SΔq. Simulating procedure was taken two times: for
16 to 222 and for 65 to 222 spiral rolls (the step of change was smaller)
Rys. 7. Wyniki testu – parametr hybrydowy SΔq. Symulacja wykonana dwa razy: dla 16÷222 i dla
65÷222 zwojów spirali (krok próbkowania był mniejszy)
4. CONCLUSION
Basing on this result of the simulating test some number of conclusions can
be underlined:
– for each analyzing surfaces, it can be possible to observe the stabilize point
or some trends (probably linear)
– the parameters from spiral grids have difference of values, but sometimes
they become constant
– it is further possible to make the spiral measurement accurate by using table
corrections for topography parameters (by define the difference between
parameters obtained from rectangular grid and spiral grid, or by define the shift
principle)
– the table corrections must be produced separately for each kind of surfaces
– in further research it is necessary to pay more attention to spatial
parameters
The next step to understand and characterize the difference between the
three-dimensional surfaces parameter, which are based on a rectangular grid and
spiral grid, will be confirmation of observed trends by repeating simulation
procedure for more as one surfaces of particular kind of machining. After that,
all result must be experimental confirm.
80
R. Majchrowski
If difference of sampling based on spiral for surface topography will be good
determine, it make possible to take this method into industry and widen the
topography assessment by means of stylus profilometers.
REFERENCES
[1] Cellary A., Wieczorowski M., Siatki próbkowania w pomiarach topografii powierzchni, in:
Materiały Krajowego Kongresu Metrologii: „Nowe wyzwania i wizje metrologii”, Gdańsk
1998, vol. 4, p. 71–76.
[2] Majchrowski R., Pomiar topografii powierzchni metodą próbkowania spiralnego, in: V
International Scientific – Technical Conference for PhD Students, Automation and CA
Systems in Technology Planning and in Manufacturing, Giewartów 2004, p. 139–143.
[3] Mathia T., Zahouani H., The development of methods for characterisation of rougness in
three dimensions, ECSC-EEC-EAEC, Brussels-Luxembourg and Authors 1993.
[4] Thomas. T. R., Rough Surfaces, Second Edition, Imperial College Press 1999.
[5] Whitehouse D., Surfaces and their Measurement, Hermes Penton Science 2002.
[6] Wieczorowski M., Sampling on aspiral for surface topography, in: Metrology and
Measurment Systems, vol. 9, Warszawa, PWN 2001, p. 195–208.
[7] Wieczorowski M., Spiral sampling as a fast way of data acquisition in surface topography,
International Journal of Machine Tools & Manufacture, 2001, 41, p. 2017–2022.
[8] Zalewski A., Cegła R., Matlab – obliczenia numeryczne i ich zastosowania, Poznań,
Wydawnictwo Nakom 1997.
[9] Zahouani H., Assoul M., Vargiolu R., Mathia T., The morphological tree transform of
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Praca wpłynęła do Redakcji 29.03.2007
Recenzent: prof. dr hab. inż. Bogdan Nowicki
WPŁYW PRÓBKOWANIA SPIRALNEGO
NA WYBRANE PARAMETRY PRZESTRZENNE – ANALIZA SYMULACYJNA
Streszczenie
W artykule autor zaprezentowł rezultaty symulacji pomiaru topografii powierzchni z użyciem
próbkowania spiralnego. Większość przyrządów do pomiaru topografii powierzchni oparta jest na
pomiarze równoległych profili. Nowatorski profilometr wykorzystujący próbkowanie spiralne jest
odpowiedzią na czasochłonność pomiaru stereometrii powierzchni. Prezentowane rezultaty porównania
parametrów przestrzennych uzyskanych z pomiaru topografii z wykorzystaniem siatki prostokątnej i
spiralnej są pierwszym krokiem do określenia i scharakteryzowania wpływu próbkowania spiralnego na
parametry 3D. W przyszłości pomoże to we wprowadzeniu tej nowej metody do przemysłu jako metody
szybkiego pomiaru topografii powierzchni przyrządem stykowym.
Słowa kluczowe: próbkowanie spiralne, pomiary topografii powierzchni